Summary of "OSCILLATIONS IN 1 SHOT | Physics | Class12th | Maharashtra Board"
Summary of the Video: "OSCILLATIONS IN 1 SHOT | Physics | Class12th | Maharashtra Board"
Main Ideas and Concepts Covered:
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Introduction to Oscillations and Periodic Motion:
- Oscillation is a repetitive to-and-fro motion of a particle along the same path.
- Periodic motion repeats itself after a fixed time interval called the period (T).
- Frequency (f) is the number of oscillations per second.
- Mean position (equilibrium) and extreme positions (maximum displacement) are defined.
- Oscillations can be linear (straight line) or angular (arc of a circle).
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Simple Harmonic Motion (SHM):
- SHM is a special type of linear periodic motion where the restoring force is proportional and opposite to displacement.
- Mathematical expression for restoring force: \( F = -kx \), where \( k \) is a constant.
- Differential equation of SHM: \(\frac{d^2x}{dt^2} + \omega^2 x = 0\), where \(\omega = \sqrt{\frac{k}{m}}\) is the angular frequency.
- Velocity and acceleration expressions derived from the differential equation.
- Displacement formula in SHM: \(x = A \sin(\omega t + \phi)\), where \(A\) is amplitude and \(\phi\) is the initial phase.
- Extreme values of displacement, velocity, and acceleration:
- Displacement max = \( \pm A \)
- Velocity max = \( \pm \omega A \)
- Acceleration max = \( \pm \omega^2 A \)
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Reference Circle Method:
- SHM can be visualized as the projection of uniform circular motion on a diameter.
- The position of the particle in SHM corresponds to the shadow (projection) of a point moving in a circle.
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Phase and Initial Phase:
- Phase indicates the state of vibration at any instant.
- Initial phase \(\phi\) depends on the starting position of the oscillation.
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Energy in SHM:
- Total energy = Kinetic Energy (KE) + Potential Energy (PE).
- KE = \( \frac{1}{2} k (A^2 - x^2) \)
- PE = \( \frac{1}{2} k x^2 \)
- Total energy remains constant: \( \frac{1}{2} k A^2 \).
- KE is maximum at mean position; PE is maximum at extreme positions.
- Energy transformation between KE and PE occurs during oscillation.
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Simple Pendulum:
- Defined as a heavy bob suspended from a rigid support by a light, inextensible string.
- For small angles, restoring force \( F = mg \sin \theta \approx mg \theta \).
- Time period of Simple Pendulum: \(T = 2\pi \sqrt{\frac{l}{g}}\)
- Second pendulum: a Simple Pendulum with a time period of 2 seconds.
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Angular SHM:
- Angular displacement \(\theta\) and torque \(\tau\) are related.
- Torque is proportional and opposite to angular displacement: \(\tau = -c \theta\)
- Angular SHM differential equation analogous to linear SHM.
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Damped Oscillations:
- Oscillations with gradually decreasing amplitude.
- Amplitude reduces over time until motion stops.
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Forced Oscillations and Resonance:
- Forced Oscillations occur when an external periodic force acts on the system.
- Natural frequency is the frequency at which the system oscillates freely.
- Resonance occurs when the frequency of the external force matches the natural frequency, causing maximum amplitude.
- Examples and explanation of Resonance phenomena.
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Composition of Two SHMs:
- Resultant SHM from two SHMs of the same frequency but different amplitudes and phases.
- Resultant amplitude and phase can be found using trigonometric identities.
- Special cases where amplitudes and phases are equal or opposite.
Methodologies and Important Formulas:
- Restoring Force in SHM: \( F = -kx \)
- Differential Equation of SHM: \( \frac{d^2x}{dt^2} + \omega^2 x = 0 \) \( \omega = \sqrt{\frac{k}{m}} \)
- Displacement in SHM: \( x = A \sin(\omega t + \phi) \)
- Velocity in SH
Category
Educational
